Unfortunately, the categories \((\Delta _{\le 2})^{op}\)  and \(((\Delta _{\le 2})^{op})^{\rhd }\)  are not cofibrant in Lack's model structure on 2-categories. Indeed the generating morphisms satisfy non-trivial relations (cf. lemma:trunc simplex gen). Hence our first task is to find appropriate cofibrant replacements, for which we apply the theory of computads introduced in [1]}. A computad consists of a graph \({G}\)  together with a set of 2-arrows between parallel morphisms in the free category on \({G}\) . We refer to the original source [1]} for a more complete description, as we need only certain computads.
